An arithmetical theorem for partially ordered sets

F 0 R P A R T I A L L Y ORDERED S E T S

Thesis by

Worthie Doyle

In Partial Fulfillment of the Requirements For the Degree of

Doctor of Philosophy

California Institute of Technology Pasadena, California

1950

I thank Professor R. P. Dilworth for the suggestion leading to this thesis, for his guidance in carrying it out, and for general patience and kindness beyond the call of duty.

The simplest multiplicative systems in which arithmetical ideas can be defined are semigroups. For such systems irreduc- ible (prime) elements can be introduced and conditions under which the fundamental theorem of arithmetic holds have been in- vestigated (Clifford (3)). After identifying associates>the el- ements of the semigroup form a partially ordered set with respect to the ordinary division relation. This suggests the possibility of an analogous arithmetical result for abstract partially or- dered sets. Although nothing corresponding to product exists in a partially ordered set, there is a notion similar to g.c.d.

This is the meet operation, defined as greatest lower bound.

Thus irreducible elements, namely those elements not expressible as meets of proper divisors can be introduced. The assumption of the ascending chain condition then implies that each element

is representable as a reduced meet of irreducibles. The central problem of this thesis is to determine conditions on the struc- ture of the partially ordered set in order that each element have a unique such representation.

Part I contains preliminary results and introduces the prin- cipal tools of the investigation. In the second part, basic pro- perties of the lattice of ideals and the connection between its structure and the irreducible decompositions of elements are de- veloped. The proofs of these results are identical with the cor- responding ones for the lattice case (Dilworth (2)). The last part contains those results whose proofs are peculiar to partial- ly ordered sets and also contains the proof of the main theorem.

PART TITLE PAGE

I Definitions and Preliminary Results. • • • 1 II Upper Semi-modular Partially Ordered Sets. 9 III Partially Ordered Sets with Unique Decomp-

ositions • References

. . . . . . . . . .

. . .

15 25

DEFINITIONS AND PRELIMINARY RESULTS

A set S is partially ordered by the inclusion relation 2 if for all x, y and z in S

(1) X ~ X

(2) X '2 y andy:! x imply y : ^X

(3) X"2. y and y -a z imply x '2.. z.

Proper inclusion will be denoted by ^~ •

Definition 1.1 Given a set of elements a, , ••. , a" inS, if an element a exists in S such that

(1)

a. _... 2 a for all i and

(2) a._ -=> b for all i implies a ::> b, then a will be called

the meet of the ai. The meet if it exists is unique and will be written a

=

^a _I ¹¹ • • • " a _1'1 or a •ll _i. a-_&. •

Lemma 1.1

Clearly a ~ B:.-... for all i and j. If b C a.

.,,

- for all i and J, then b ~ a~ for all i and hence b ~ a.

This is a sort of associative law for meets.

Definition 1.2 If a : a, ^A • • • " " a., always implies a : a, for some i, then a will be called (meet) irreducible.

Theorem 1.1 If S satisfies the ascending chain condition, then every element of S is representable as a meet of irreduc- ibles.

A maximal element is already irreducible. Let the theorem hold for all proper divisors of a. If a is irreducible, a is

the desired representation; otherwise there exist a~ so a =~ aL ,

where a1^~ a. Then the induction hypothesis implies irreducibles q __ exist so that a. =~ Cl- for each i. By lemma 1.1, a : 1) q ..

.. j .... J J ~; :4j

and the theorem follows by ~~~n1 for all a in

s.

Definition 1.3 A subset A of S will be called an ideal (dual) if

(1) x c: A and y 2 x imply y ^£: A and (2) x : /) _.t x. and x. ^E A imply x e A.

A-

For any element x of S the set of elements y such that

y ~ xis an ideal called the principal ideal generated by x and denoted by (x). The set of ideals of S forms a partially ordered set, L, under the relation A2 B if and only if A is a subset of B. The inclusion relation has been inverted so that

(x) -::> {y) in L if and only if x 2 y in

s.

Lemma 1.2 ^L is a complete lattice.

Given a set of ideals A~ of L, their set meet is evidently an ideal and furnishes their lattice union ~ Ar• L also has a null element, the ideal S itself. Hence (Birkhoff (1) ch. 4, thm.2) L is a complete lattice.

If A, 2 ••• ? A"?. ••• , then

Q

^{Acr is the set} join of the A_.. Corollary 1.1 (a) =

0

^{a"-) if and only if a

= Q

^a.._.

If (a) =()

...

(a..L), then (a₄ ) 2 (a) and hence a₄ 2 a for all i • If b £ a-' for all 1, then (b) ~ (a~) and ~b)£ 0 Ca;.) • (a) • Hence b £ a and a

= n

^{a. .} Similarly a =(l a~ implies (a) :: () (a..t).

;L • .... ..

The following lemma provides a constructive definition of ideal meet.

Lemma 1.3 ^{~ A¥} may be defined inductively as the set union

"I<

of S₀ ^{, • • • ,} S", ••• , where So is the set union of the A, and S ^I<

consists of all elements of S containing meets of elements of

By the definition of ideal and ideal inclusion nAv must

f/

contain at least all the elements mentioned, and these clearly form an ideal.

Arguments will frequently be made using induction on the index k of S~ Hence the following obvious sharpening of lemma 1.3 will be convenient.

Lemma 1.4 If S is a partially ordered set in which every element is expressible as a meet of irreducibles, then nAy is

v

the set union 8₀ ^{, • • • ,} SK, ••• ,where So is the set union of the Avand SK consists of all elements of S containing meets of irreducibles of SK_₁•

If a ^E S~, then a 2 b : f} a..._ for a""' e: S.:-•. If a_... . = I) q_{:1 ..} . . _., for irreducibles q..:j' then the definition of sk-I implies ~j ^E SK-\

while lemma 1.1 implies a 2 b : 1\q ..•

.:._; "'".J

Theorem 1.2 If (a) =(\A. then a : fl a .. where a .. ~ A; •

A. .... 4; ^A-J "-l -

Let S be the classes of lemma 1.3 for the meet n_;. A. and

...

suppose a E- s. for k > 1. Then a "2 l\ a. for a.: • Sw_,. Since

A. ... - "'

each a...._ 4i {a) it follows that a :

0 ...

a.._. Since a₄ e SK-₁ there are b₄j tt SK-t such that for each i

since every b .. 2 a, wbile if b C£:

"'j

a_-.-;1 /lb. . _{-i ~J} • Then a

=

_{•.; - ,}

11

^{b .. c S} _lt-J

b.._.i for all i and j then

b £ a. for all i and b~

l.l a.:

: a . After k-1 such steps it fol-

.... A- ~

lows that a a S₁ or a :._{.... ,} (\a_{• j} .. with a .. EA ••

'"j Ao

In most applications of theorem 1.2 every element of S will be representable as a meet of irreducibles. In this case the elements a4j may be taken to be irreducible.

Lemma 1.5 (q) is irreducible in L if and only if q is ir- reducible in

s.

If q =O

...

x. for ^~ x

...

.=> q then corollary 1.1 implies (q) =fl^;.. (x"' .) for (x._.... ) ~ (q). Conversely suppose q is irreducible and let

( q) : A ⁿ B. Then theorem 1. 2 implies there exist aA.. .s A, b• c B such that q :. _{a "' ••• "' a "'b " ••• "'b .}

' K I n By irreducibility of q either q:: a-4. and (q ) ? A or q • b;. and (q) -=> B. Hence (q)= A or (q) : B, and (q) is irreducible.

Lemma 1.6 Let every element of S be expressible as a meet of irreducibles. Then A ^~ B implies there exists an irreduc-

ible q such that q ^€ B but q ~ A.

Let b ^t- B, b ; A and b :: l.l q_:. for irreducibles q. . Since

""

"'

b f. A, there is an i so q. _~

tl

A, and q. _~ ² b implies q. _~ ^E B.

Lemma 1.7 Let ll₁ ^{'2. • • • :2 M,}

2. ...

be a chain of ideals of L such that every M""=>{a) an~ P =QM,.. Then P ~(tk),

For if P

=

(a), then a ^E- M"l" for some 't', contradicting

M't => (a).

In any partially ordered set x is said to cover y (written x ~ y) if x ~ y while no z exists for which x ~ z J y.

Theorem 1.3 Let B :::> (a) in L. Then there exists an ideal P such that B 2 P > (a).

This follows by applying lemma 1.7 and the maximum princi- ple to the set of ideals N such that B 2 N ::> (a).

Definition 1.4 An ideal A will be called upper semi-modu- lar in L if B 2 A, C >- A and B ;j. C imply B >~ C ~ B.

The partially ordered set S will be called upper semi-mod- ular if (x) is upper semi-modular in L for every x in

s.

The following four lemmas concern a single upper semi-mod- ular element of a lattice.

Lemma 1.8 Let A be an upper semi-modular element of a lat- tice L and A, , ••• , An> A. Then each union independent set of the A~ is contained in a maximal independent set. The union of the A~ in a maximal independent set contains every A~.

Let M =(A,, •.• , Ar) be the given union independent set.

Since there is only a finite number of subsets of A, ••• An

containing A, ••• A~ a maximal independent set containing A, ••• Ar exists.

Let (A₁, • • • , A₅ ⁾ be maximal union independent. Suppose

I -

Let A. _A. - A_• ., ••• vA_.-.-1 . " A."· •• _... vA , _s 1 ~ i ~ $. Since (A, , ••• , A₅ ^, AK) is dependent there is an i so

I I

By upper semi-modularity A;_" A"'>- A;.. and Hence A _~ I . .., A _IC.

=

^{U A}_-^-~A _K ^•

Lemma 1.9 Let A be an upper semi-modular element of a lat- tice land A ₁^{, • • • ,} A$>- Ao• Then each union independent set of the A~ generates a Boolean algebra.

Let A and B be two subsets of the union independent set (A,, ..• ,An) and ~A denote the union of the elements of A.

Let Au B and An B denote set union and set meet of the sets A and B. Evidently (~A) u (~B) = ~(A u B), while ~A = ^~B implies A= B by the independence of the AL, o ~ i ~ s . Next it is shown that

(1) (U) ~ (~B) : ~(A" B).

Let m(A) denote the number of elements in A and n(A)= n-m(A).

If n(A" B) : 0, then m(A/\ B) : n and A: B. If n(A, B) : 1, either A 2 B or B 2 A and (1) holds. Let (1) hold for all A

and B such that n(A" B)<! and suppose n(An B) • 1 for some A

and B. Then m(A" B)

=

^{n-1. •} r so that A: (A, , ••. , A.,.., A,..""', .•• ,At:) and B

=

^(A1 , • • • A__.. , A' _{~ .. ,} , ••• , A¹_t: .) . Since {1) is trivial if B2 A

i t may be assumed that t > r. ~).

Now m{A" B')

=

^{r • 1} and hence n(A" B') ::. .P-- 1 < 1.. By the induc- t ion assumption ~(A,.. B ^{1 )} = (~A) ⁿ (fi'). Hence ~(A ~ B')

=

^{('iA) n}

{~B') 2.. (u) ~ (n)2 ~(An B). Since A,, ••• , An are independent, l:{A, B)~ A ... and hence ~(An B') >-~A n B). If ~(A" B') : (l:A) ~

(~B) then ~B 2 Ah,, contradicting independence of A,, ••• , A"'.

Renee (~A)n (~B) • z (A, B) and (1) holds for n(A, B) =~. By induction (1) holds for all A and B.

Thus the elements expressible as joins of A, , ••• , An are isomorphic to the subsets of (A,, ••• , An) under meet and join and A₁, • • • , An generate a Boolean algebra.

Lemma 1.10 Let A be an upper semi-modular element of a lat- tice L and A, , ••• , Ah}- A. Then any two maximal union inde:J:e n- dent sets of the A~ have the same number of elements andany ele- ment of one set may be replaced by a suitably chosen element of the other without altering the maximal property.

By lemma 1.8 if A: ••• A~ and

A : ...

^A~ are two maximal inde- pendent sets of the A;.. then UA~ ^: VA~. ^{If every} A~ ^£B =JJf~,

then B 2 U A~' ^'2 A~ contradicts independence of A~. Hence there exists an

t ;

such that B

1

A~'. Then UA; "2 B ^u A; ~ B. by upper

" I I I I

semi-modularity of A, and B v A1 = ^UA₄ ^• Hence A, , ••• ,A,_, ,Ai.~u

A' A" 1.· f . d d t . 1 i d d t

••• , K' ~ ' 1.n epen en are max1.ma n epen en. If not

•• I ,. '

independent, since B '2 A-i there exists an A_. such that C u A;, 2 A,_

U ^I VA. •• .. t . ^{I I I}

where

c :

^Ah. Th.cn ^j -=:SuA.= (uA.l ~ C '~" .-a.ltc.i:s UA. > ^UAtt }-U Ah.

h~ ... yi. ³ s h~A. h • ..., ..

The replacement property implies k : 1.. For if k

<.t ,

after

I I I

replacing every A~ by an A~ there would remain some

A ;

^divisible

by the union of the _A~' that I

3 replaced the A~ , contradicting the 1 d d Of the A,_,.

n epen ence ~ Similarly t~ k, and ! : k.

Lemma 1.11 Let A be an upper semi-modular element of a lat- tice and A, ••• AK>A. Then any chain joining VA;.. to A has not more than k • 1 distinct members.

Clearly A, ••• AK may be supposed independent. Let

A :: B₀ c B, c ••• c B.~,. : UA.:. and suppose .L~k. By upper semi-mod- ularity B_o ...(. B _o u A₁< ••• < B _o u A _l u ••• uA _K-1 <A _I v ••• uA. _K Suppose it has been shown that B c B c. ••• c..B. < B. ^u A,-<. ••• -<. B- u A ,v •• • v A

D I ~ - ~ ~~

where k. s k-1 and i < k.

4 Consider the chain

B0 C B, ^{C • • . C} B.L•t ~B..i.+t v Aa ~ • • • ~ BL.~, v A,v o • • v AIC.-I !;A, u • • • v A.,.

~

and assume all its members are distinct. If B"-" A,v ••• vAK--•~B'"'-~''

4

}B.v A,u ••• uA and hence A u ••• vA._ : B. v A,v •• • v AK ,contrary

A. K..L-1 I .... "-i-l .,:.-

to assumption. Thus Bk"' A,u ••• v AK __ ₁2B ... ,. Continuing in this ...

way, eventually B~" A₁ 2. B..,L+t• But then B.t ^u A, 2 B ... -:. B.A. and

BA. v A,> B..&. imply B..L" A,

=

^{B;.'"• BA.+, .. A}1 , contradicting the assumP-

tion. Hence at least two members of the chain considered are equal. Thus (renumbering the A;_ if necessary) B. c B, ~ •••

c B~ c. B. ~B. " A -< •••

-< B .:_.,

^{v A} ₁ ^{v . . . ... A_. <A}₁ ^{• • • • u A , where}

- ... &.+1 ' - . .. Lt'- · C ~

ki.+,~ k..._ -1 ~ k -(i + 1). By induction B. c ..• c B .. _,-<A,"••· ^u Aec:

where r ~ k. But then B.,..= A, ^{u • • •} ~A"'- and hence r • J. , contra- dicting .t>k. Thus ..l~k and the lemma follows.

PART II

UPPER SEMI-MODULAR PARTIALLY ORDERED SETS

This section contains some general theorems relating the irreducible decompositions of elements of S to structural prop- erties of the lattice of ideals L• All the proofs of these theorems are those of Dilworth (2).

Let U~denote the union of all ideals covering (a) in L and let L~ denote the quotient lattice U~f(a). Lemma 1.11 than implies.

Lemma 2.1 Let S be upper semi-modular. Then if P, ••• PK is a maximal independent set of point ideals of ^{L~ the} length of any chain of La, is at most k.

According to lemma 2.1, La, is Archimedean if and only if

U~ is the Join of a finite number of point ideals of L~.

Theorem 2.1 Let S be upper semi-modular and every element be expressible as a meet of irreducibles. Then La, is Archi- medean if and only if the number of components in the reduced irreducible decompositions of a is bounded.

(A) If ^L~ is not Archimedean let n be any integer. Then lemma 2.1 implies there exist n union independent ^P~ covering

(a). By lemma 1.9 these generate a Boolean algebra. Let

A;... :

. V.

^{Pi , then (a)}

= 0

^A;., is a reduced representation (a in no

~;o...

meet of fewer A~). By theorem 1.2 there are irreducibles q~~ in

A~ so a •

n

^q... After eliminating superfluous factors a reduced

.., ..:..,~ .... 1

representation having at least n irreducible components is had,

since unless every A;... is represented a would be in a meet of fewer AL. Hence the number of components in representations of a is unbounded.

(B) Let the number of components be unbounded so that for each k there exists a reduced decomposition into irreducibles, a • q,"••• " qn with n ~ k. Then by lemma 1.5 and corollary 1.1

(a) •

(q,) n •.•

~ (q~) is a reduced decomposition into irreducibles in L. Let Q~

: A

^{(q- ). Then Q~~(a)} and theorem 1.3 implies

"'" ~~.... j •

there are P: so Q~ ~ P!

>

(a). If U P. 2. P, , then since (q.) ~ Q~

- - - 1~ .... 1 .... ... ""l

for all J ¢ i it follows that (q;_) 2 P~ for all J ~ i and hence (a)

= (

^{q.) "} _~ ^Q_... ^{~? P,}

-

^>- (a), a contradiction. Hence the p_ ^~ are in- dependent, for every k there exist at least k independent points of ^L~ and L.., is not Archimedean.

Theorem 2.2 Let S be upper semi-modular. Then if ^L~ is Archimedean it is complemented and every ideal of ^L~ is expres- sible as a meet of maximal ideals.

Let A be in ^L~ and P, ••• PK be a maximal independent set of points of ^L~ divisible by A. Extend P, ••• ^P~ to a maximal independent set P₁ ^{• • •} P~·

A ^v A' ² Vo.. and A ^v A ^{1 •} Uo..•

Let A ' : P _k+' v ••• v p . Then

W\

Suppose A "A' ~ (a). Then by the- oreml.3, there is a P so A " A' 2 P >- (a). Since A 2 P the max- imal property of P, ••• PK implies P, ^{v • • • v} PK ~ P by lemma 1.8.

Since by lemma 1.9 the P, ••• Pn generate a Boolean algebra, (a) : (P, ^{v • • • v} P 'It) (\ A' ~ P >-(a), a contradiction. Hence

A" A'

=

(a) and L ~is complemented.

Now let Q be irreducible in L.,.: Let P, ••• P" be a maximal independent set of points of L~divisible by Q and let this set

be extended to the maximal independent set P, ••• ^P~· Then

Q,

:t

^P~~: .. ,, ••• , P"' and hence by upper semi-modularity Q, ^v P;... >- Q

for i ^-.s k .., 1, ••• , n. Since Q, is irreducible Q. ..., P~ .. ,:. ••• ^=.

Q,.., P ,· hence _\"\ U .. _{_.}

=

ⁿ _... ^.., Uo...:. Q, .. P _'K+\ "'• •• .., P : _h ^Q"' P _K+l ,.. ^Q and Q is maximal in ^L~· Since L& is Archimedean every ideal is expres-

sible as a meet of irreducibles of L~ and hence as a meet of maximal ideals of Lo...

If L.., is Archimedean an arbitrary complement of A in La., will be denoted by A'.

In the -following a connection is established between the irreducible representations of a in S and certain representa- tions of (a) in L~.

Definition 2.1 An ideal C;:. Va. of Lo- is called characteris- tic if there is an irreducible q of S dividing exactly the same point ideals of L~ as

c.

Theorem 2.3 An element a of S has a reduced representation

a

= ⁰

_.... ^qk for irreducibles q~ if and only if (a) has a reduced

representation (a)

: 0.

^{C,.:. ,} where ^C.A. are characteristic ideals

...

of Lta.. such that q..._ e c"'-.

(A) Let a :

n

^q. be a reduced representation with irre-

'"' N

ducibles q.L. If any q. 6 Vo.- then ^Q.!

= (\ (

^{q. )} ~ (a) by corollary

" ' A. j~A. ~

1.1 since the representation is reduced. Hence there exists a P;,. such that Q.l 2 PA. >- (a) and (a)

= (q.J "

Ql ² P..._ >- (a) is a contradiction. So q.

t

^{Uo..• Let C. ~ (q. )} be a characteristic

~ • N

ideal of L~ associated with q~. One is always to hand since the union of those points of Lta.. divisible by (q~) will serve

and is different from Ua. since q~ tj UOv • Then (a) : (\ (q.) 2

.:. ""

n

_.A.

c.

_.,... ² {a) implies (a)~

A

_{,._} ^c~. If this is not a reduced repre-

sentation,

c.

^{2. (\}

c_.

and ^Q! 2 P. >- (a) together with the defini-

6> j'f:...t .J A. ...

tion of charl'deristic ideal imply (a) : () Ci :=!. P. >- (a), a con-

~~)., ? " "

tradiction. Hence the representation is reduced.

(B) Let (a) =

0

_.,., ^C~ be a reduced representation with as- sociated irreducibles q. ^E CA.. If (\ (q.) # {a) there's a P so

" ' .4 .Jw

0 Cq"-) :2 P >(a) and hence (a) :

0

^{C;._ ~ P .. >-} (a), a contradiction.

Hence a= ~ q .• If this is not reduced, part A of the proof

A. " '

implies

f:.

_,.. C;... is not reduced.

The next theorem gives a characterization of characteristic ideals in terms of the structure of ^L~·

Theorem 2. 4 Let S be upper semi-modular and each element ex- pressible as a meet of irreducibles. Then if L~ is Archimedean C is characteristic if and only if there exists an ideal R of L such that R 2 C,

c• v

^R>R and C'" R= (a) for every

c•.

(A) Let such an ideal exist. Then V~v R ::. C ^v C ^1"' R :. C' ... R.

By lemma 1. 6 there's an irreducible q so q e R, q 4:- C' ^u R

=

^{U,.. v R.}

Since ( q)

=

^{R ~} C, ( q) divides every point ideal of Lo.. that C does.

On the other hand let (q) ~ P >- (a). Then if R ~ P, C^{1 v} R:

Uo..., R ~ P u R ^=> R. Hence C' ^v R :. P ^v R and ( q) ² P " R : C'.., R :

v ...

^u R, contradicting the choice of q. Hence R a. P. If C ~ P,

then

c•

^a P for some

c•,

and (a)= C'"" R ² P >-{a), which is im- possible. Hence (q)

=

P implies C ~ P and C is characteristic.

(B) Let C be characteristic and q an associated irreducible;

(q) will be shown to have the properties required of R. Since

C

I!

Ua., C' #: (a) and hence P exists so C ^{1 2} P >- (a). Since q is irreducible (lemma 1.5), (q) ^v P :: (q) .. Uo..• Hence (q)" V-.,=

(q) u 0':: (q) v P >- (q). If 0'" (q) -:F. (a) there is a P so

0¹"" (q) ~ P>- (a) and hence 0'" C ::! P>- (a) since Cis character-

istic associated with P. This is impossible; hence c• ~ (q) :(a).

Corollary 2.1 Each maximal ideal of L ^o.... is characteristic.

For the R of theorem 2.4 the maximal ideal itself may be taken.

Theorem 2.5 Let S be upper semi-modular and every ele•ent be expressible as a meet of irreducibles. Then if L~ is Archime- dean each characteristic ideal C of L~ occurs in a reduced rep- resentation (a)= C"

c, ... ...

"" C.c, where k is the number of max- imal independent point ideals of ^{L~ divisible} by C and Ck are characteristic ideals of La•

Let P, ••• Pte be a maximal independent set of point ideals divisible by C and imbed them in a maximal independent set

P, ••• P ..._.. . Let C. =_{,._ ~.,...} U p. ₁ fori= 1, ••• , k. If C" C," ••• ,..Ott* (a), there is a P so C " C," ••• " CK ² P >(a) and C ² P implies

p, ^{v • • • v} P" ~ P by lemma 1. 8. Since P, ••• P'"' generate a Boolean algebra (a)

=

(P,"' ••• "' Pl() " C, "'• •• " C~<- 2 P >- (a), a contradiction.

Hence (a)

= c" c," ...

^{" OK.} Since

c

^{1\ 0,1\}

P. _... ~ (a) the representation is reduced.

deals of Lo.. they are characteristic.

...

^"

Since C. are maximal i-_{4 .}

Corollary 2.2 Let S be upper semi-modular and every element be expressible as a meet of irreducibles. Then if L ^0.; is Archi-

medean of length k, a has a reduced decomposition into irreduc- ibles with k components.

By lemma 1.9, theorem 2.5, and theorem 2.4.

PART III

PARTIALLY ORDERED SETS WITH UNIQUE DECOMPOSITIONS

The object of this section is to characterise those par- tially ordered sets for which every element has a unique irre- ducible decomposition. The main result is

Theorem 3.1 • Let S satisfy the ascending chain condition.

Then each element of S has a unique representation as a reduced meet of irreducibles if and only if S is upper semi-modular and Lo.. is a Boolean algebra for each a.

This theorem will follow from a series of lemmas the first of which proves the necessity of these conditions.

Lemma 3.1 Let S satisfy the ascending chain condition and each element of S have a unique representation as a reduced meet of irreduoibles. Then S is upper semi-modular and Lo..is a Boolean algebra for each a.

(A) Let B >- (.a), C ^i!! (a) , C ~ B and suppose there is a D

so B ^u C ^=> D =>

c.

By lemma 1.6 there are irreducibles qc:. and q.l)

inS such that qce C, qc:.j D, q~e D and q3>4 B"" C (hence qn¢ B).

Now B ² B "' (qe) 2 (a). If B : B" (qc.) then q, E B, and qc:. ^e B" ^{C : l D} implies qcE D, a contradiction. Hence (a) • B ~ (qc). Also

B ^::> B" (q.J 2 (a) and if B

=

^{B "} (q ) then q e B, a contradiction;

- 1) J)

so (a)

=

^{B" (q))).}

By theorem 1.2 there are irreducibles b., b_{... ...} I . in B and c. , _....

d~ in (qc), (qD) respectively such that a

=

^b I ^{" •••"b} ""' ^{1'\ c} I ^"-••

I I

b, " • • • " bK " d₁ 1\ • • • 1\ d.t • If y ~ b"'- for all i and y £ qc

then y ^~ a. Hence a :: b _I " ••• " b _too\ " q,._... . Similarly

- b' b'

a - I " • • • 1\ K f\ qJ). Both representations may be assumed reduced.

Unless qc actually occurs after converting to a reduced repre- sentation it will follow that a =

0

b..;. e B >- (a), which is impos- sible. Similarly q» actually occurs in the second representa- tion. If qc

=

ql) then qc. is in D, while if qc. • b.L^I , then a ^E B >- (a), a contradiction in either case. Hence the two rep- resentations are distinct. This contradicts uniqueness; hence

B ^v C >-C, and S is upper semi-modular.

The necessity of upper semi-modularity in the lattice case was first noticed by Morgan Ward.

(B) Since the decomposition is unique the number of com- ponents is bounded for any a and L .,__ is Archimedean by Theorem 2.1. Let P, ••• PK be a maximal independent set of point ideals of Lo.. • By lemma 1. 9 they generate a Boolean algebra having maximal ideals M ₁ ^{• • •} M\C. Since ^\Jo.

= ^\i

^P""'> M.._, the M:... are also maximal ideals of ^L~ and hence are characteristic by corollary Since {a)

=

⁰_"' ^M_.... ^. , a has a reduced decomposition into ir- reducibles, ^{a ::} (\ q_{..., -}, , with q_:.4. . ^E M .•

A. Let M be any other maximal ideal of L... Then there is an irreducible q £ M so that q f/: V~.

M is a characteristic ideal associated with the irreducible q.

By theorems 2.5 and 2.3 q is a component in some decomposition of a. Since the decompositions are assumed unique q

=

^{qA for}

some i, and q ^E. M ^v M;.. • ^v~ is a contradiction unless M

=

MA..•

Hence ^M~ are all the maximal ideals of ^L~, and by theorem 2.2 all the elements of L~are in the Boolean algebra generated by the P.4. .

Lemma 3.2 If L"' is a Boolean algebra it is Archimedean.

If L ^Q.. has an infinite number of point ideals let P, • • • p"' •••

be a denumerable sequence of them and define P;

= (\

P. • Since

,_ i:I:A. ^A.

Lo.. is a Boolean algebra, (a) • D _... P;. Because an infinite meet of ideals consists of all elements in finite meets (a) ^{: p I t\ • • •}

~P~ for some k. Then (a) ~ P~+,> {a), a contradiction; hence

'

L~ has only a finite number of point ideals and is Archimedean by lemma 2.1.

Lemma 3.3 Let S be upper semi-modular and satisfy the as- cending chain condition. Let every three ideals covering a principal ideal generate a Boolean algebra and let q be an ir- reducible of S such that q ² a, B and C ,... (a) and B :F

c.

Then either q ^E: B or q e

c.

The lemma is proved by a double induction, the first on the element a using the ascending chain condition. The lemma holds vacuously if a is maximal. Assume it holds for all prop- er divisors of a in

a.

Since B" C = {a), the irreducible q is distinct from a. Hence by theorem 1.3 there exists an ideal A such that {q) ² A ^)o {a). If A :f=. B, C the hypothesis of the lemma implies A

=

(A.., B) " {A" C), while if A • B or A

=

^{C the}

conclusion is trivial.

It is convenent to prove the lemma for all irreducibles q' in A. Let S₀ ^{• • •} S~ ••• be the classes of lemma 1.4 for the meet of A.., B and A"

c.

The second induction will be on the index k of SK. If q ^{1 E} S₀ ^, then q ^{1 E} A " B ~ B or q ^{1 E} A.., C ² C and the lemma holds. Now suppose the lemma is true for all q IE s\C:. for k < n and let q' ^E s\'\' q. f. sl\_,. By lemma 1. ~ there

exist irreducibles q~ E S~-• such that q ¹ ~ y : q\" ••• " q.._ , where the representation of y is reduced.

If I : 1 then q'E Sh_,, hence 1..>1. Then each q..._ e: A and the induction hypothats on SK implies that for each i either qA£ B or qA-

e c.

^{If {y) v}

c ..

^{{y) u B, then q~ E B v}

c

^{for all i and}

hence {q') 2 (y) 2 Bv C gives the lemma for q'. If (y)" C =F {y) v B, by upper semi-modularity both B¹ : {y)v Band C' :(y) v C cover (y). Since (q') ~ (y) ? A>- (a) y is a proper divisor of a. Hence by the first induction q • € B ¹ ~ B or q ¹ ~ C' ^:2

c.

This proves the lemma for S~ and the induction is complete.

Lemma 3.4 Let S be upper semi-modular and satisfy the as- cending chain condition, and let every three ideals covering a principal ideal generate a Boolean algebra. If a : q\" ••• ,. q"' is a reduced decomposition into irreducibles, then L~ is Archi- medean of length k and each (q.) divides a maximal ideal of L~.

/'-

Let A;.. be the union of the point ideals of La. contained in

(q~). Then A~ is a characteristic ideal associated with the ir- reducible q;.. (theorem 2.3). Since A;_¢ V"' there is a point P such that A; ..

Ji

P. By lemma 3. 3 A;.. : P ¹ for every point P ¹ of L ._

that is different from P. Hence A;.." P : V.._and A;.. is maximal by upper semi-modularity.

Let B₀ ^:: Vo., and

a ...

be the union of points of L~ divisible by (q₁) , • • • , (q..-.). Then B, ::. A._, and B.). B,. Evidently B.c._₁2 B.t.•

If B.a.-, ::. B.._ then .f\ ( q. ) 2 P. >- (a) since representation is re-

;1*~ ~ •

duced. Then ( q\ )" ••• ^{1\ (} q~_,) ~ P..t. implies B..e. ::. B.c._, 2 P~ • This implies {qt ) 2 p.L and hence (a)~

n

_,.. ^{(q.._) :=! ~ > (a),} a contradic- tion. Hence ~-• • B~. Let P and P' be two point ideals of L~

divisible by B.t_,. Since q.t is irreducible q.L E: P or qL E- P' by lemma 3. 3. Then BL '2 P or B..t. 2 P'. Hence B.t_, : B..~- ^v P (say) and B_{1 _,} > BL by upper semi-modularity. Hence the chain Uo..."B,>

··· >

BK: (a) has length k and by lemma 2.1 LA,. is Archimedean of length k.

Lemma 3.5 Let S be upper semi-modular satisfying the as- cending chain condition and let every three ideals covering a principal ideal generate a Boolean algebra. If q and q' are irreducibles dividing a, while P is a point of L~ such that neither (q) nor (q^{1 )} divides P, then q : q'.

The proof iB by induction on a. The lemma holds vacuously if a is maximal. Let it hold for all proper divisors of a.

If P is the only point of Lea., then q :;, a implies ( q) ² P ,... (a) by theorem 1.3, a contradiction; thus q ~ a. Similarly q¹

=

^a,

and the lemma holds for a in this case. Otherwise there exists a point P' in L~ so P'-* P· Then lemma 3.3 implies q, q' e P'·

Define A

=

^{(q) "'} (q'), and let St< be the classes of lemma 1.4 for this meet. It will next be shown that q and q' are the on- ly irreducibles of A not in P. Suppose r ^E A and r ¢ P. Then if r E. SK, r 2 y : s\" ••• " an for SA. irreducibles of

sl(_, .

^Since

(y) ~ P, (y) ^v P is a point of L~ and there is an sA. so s;..

4

(y) " P.

Since (y) :? A ² P ¹ >- (a), y :;, a and the induction hypothesis implies r = s.., ^f;

S "_,.

Repeating this argument, eventually r e So. If r e (q) and r~q then Q exists such that (r):? Q >-(q). By upper semi-modularity (q) u P > (q), while (r)~ P implies (q) v P:FQ.

Thus (q) • Q" (P ^v (q)), contradicting irreducibility. Hence r = q. Similarly r e (q •) implies r • q'.

Now define A₀ ~ (q), B₀

=

^(q') and by induction

AK

=

A\(_₁^" (P"' BK_₁ ^{) ,} Bt( = B"_' "' (P "'AK-\). Note that Aw.+f (PvBK.)"

A K.

=

^{(P " BK) " (P " BK.-I)"} AK-I': • • • : (P '"' B,J • • • (P., B. ) ^1'\ (q) : (P v Bt<) " (q). Let SK be as above. It will next be shown that if x ~ P .., A and x ^E S , then x e A'tJC. or x ^e- BK.. Trivially if x e So , then x ~ A0 or x e Bo. Suppose the statement holds for all ele- menta of s~_,and let x e SK. Then x ² y ::. r,"' • • • r""' for irreduc- ibles r"" e SK_,. Two possibilities exist. If y e P, then all rA.. ^E. P ^v A and by the induction hypothesis on SK_, each r..._ is iml A v p orB v p. Hence every r;_e (A _,uP) " (B.,._,vP) 2 A~ and

1(•1 1<-1 ~ "

therefor x e AK. If y ^ft. P, then (y) ^v P >-(y). Then the truth of the lemma for y :) a implies there is exactly one r..;.., say r , , so that r, (/. P ^v (y) while r2. , ••• , r ... e P. Then the result for SK_, implies rl. , ••• , r_ e (A>\_," P) " (BK-• ^v P) while the preceding para- graph gives r ₁

=

^{q t: Ao "2 AK_, or r , :. q 1 e B}0 ~ B"_,. Thus

x e:AK_," (P " B"_,) ::. AK or x ^e Bot_,"(P .. AK_₁^{) •} B\( and the statement follows for x ^E SK.

Let C

=

0A...;,.. Since Ao 2 A,= • • • this is the set union of the A;..· Now P ^u A~

c.

For if x ~ P ^u A and x ~Sa<., then either x ^E A~ 2. C or x ^e B\<.., P ~ All(,..,~

c.

Also C ² A. For clearly A0 ,

B0 :! A while if AK, B\(. 2 A then At<.t-l

=

(B~ "P) ^1'\ A"~ A and BKH ^2. A.

By upper semi-·modulari ty P "' A>-A and hence P ^u A :. C or C ::. A.

If p ^v A

=

C then ( q) :. A0 2 C a P contradicts q ¢ P. Therefor A :. C : (\.... A ^~ . •

Since q I E A :. C there is a k such that q.' e A\(.t-l :. (q) " (PvBK) • Let T~ be the classes of lemma 1.4 for the meet of (q) and

-P ^v BK. If q' ~ TK. then q' : y : r ," ••• " r "' for irreducibles